How do I specify in the quest which artifact needs to be obtained?

Set KIND_MANUAL, and write the following in the script:

function artifact for OSH_objective()
while 1 do
* if HasArtefact("Script name of the hero") then
* * SetObjectiveState("Script name of the quest", OBJECTIVE_COMPLETED);
* * break
* end;
sleep(2)
end;
end;

startThread(artifact_objective)

It's easier to write this in the script than to mess around with setting up quests in the editor.